From cfed4cf5577c2e1d539c3533fadd128637df5166 Mon Sep 17 00:00:00 2001 From: Paul Schrimpf Date: Fri, 24 Nov 2023 12:27:29 -0800 Subject: [PATCH] ps7 --- site/626.bib | 15 ++++++++ site/problemsets/07/chen-fig1.png | Bin 0 -> 130424 bytes site/problemsets/07/chen-tab2.png | Bin 0 -> 103761 bytes site/problemsets/07/ps07.qmd | 45 +++++++++++++++++++++++- site/problemsets/07/ps07_2022.qmd | 55 ++++++++++++++++++++++++++++++ 5 files changed, 114 insertions(+), 1 deletion(-) create mode 100644 site/problemsets/07/chen-fig1.png create mode 100644 site/problemsets/07/chen-tab2.png create mode 100644 site/problemsets/07/ps07_2022.qmd diff --git a/site/626.bib b/site/626.bib index 863e109..80948a4 100644 --- a/site/626.bib +++ b/site/626.bib @@ -628,3 +628,18 @@ @article{rambachan2023 url = {https://doi.org/10.1093/restud/rdad018}, eprint = {https://academic.oup.com/restud/article-pdf/90/5/2555/51356029/rdad018.pdf}, } + +@article{chen2023, +author = {Chen, Weiwei and French, Michael T.}, +title = {Marijuana legalization and traffic fatalities revisited}, +journal = {Southern Economic Journal}, +volume = {90}, +number = {2}, +pages = {259-276}, +keywords = {marijuana legalization, medical marijuana, recreational marijuana, traffic fatalities}, +doi = {https://doi.org/10.1002/soej.12657}, +url = {https://onlinelibrary.wiley.com/doi/abs/10.1002/soej.12657}, +eprint = {https://onlinelibrary.wiley.com/doi/pdf/10.1002/soej.12657}, +abstract = {Abstract The legal landscape for marijuana in the United States has changed dramatically over the last three decades. While several studies have examined the relationship between marijuana legalization and traffic fatalities, some of the research is becoming outdated and existing evidence remains mixed. Our research revisits the topic with two updates. First, our study includes states that legalized marijuana more recently and provides updated evidence on the effects of marijuana legalization. Second, considering recent discussions about the limitations of difference-in-differences designs, we employ alternative estimators that are robust to heterogeneous and dynamic treatment effects. Overall, our alternative estimators suggest either a smaller reduction (i.e., 3.9\% drop in the overall fatality rate) or no change in traffic fatalities associated with legalizing marijuana for medical use, compared to the two-way fixed-effect estimator. We find no significant impact on traffic fatalities associated with legalizing marijuana for recreational use.}, +year = {2023} +} diff --git a/site/problemsets/07/chen-fig1.png b/site/problemsets/07/chen-fig1.png new file mode 100644 index 0000000000000000000000000000000000000000..f4b3cac775719817b5631c20262535f911d56ceb GIT binary patch literal 130424 zcmZ5|1yohh+U-#kP(Vt$Tco=~kS^&)Lb^*jltW8MHzM6F9n#$?-JOTdx48FzW4!+j zkDNWWx?`;`=QpRppX4P_UgN)pKp-emk{^{I5I8gNy9Dt$ICI~b*$%!)jU<)iAP{$I z2*ei(f!u&ozS|IpGcyFTs}F(jCO{xKb}99W{NNAZ4P_)gLLQ%f{c6gM24~QW4CKE; zAgJK*MHvM0DhwQy5b$c$fvZ5IK7LSf{kOm1p(A_!h;XdH!7f{nv$tIb5`=l&f4`Flu7+^2m zYHwc~NMth=gbA{xFK&4M8-tA_OCR0wu`(2t1W|z*&Pz{nnzQUR*F3nu9YULBefI)X@d} zp*AN%AOb`Xr6K#}2VfMnZ}15-^Rn3g_U^zDOiWG^g$N2W;NW0|;d1QC!L$j9Ay+E6 zswfarS0b(gUS`(?gfwRddwU1_jJWBW(!;6jP^%gnZhpwIc@j@bjTPrA1d?95+1amV z`L^YN?9b?x#rO#{Fknc#_LG_#_L3xcLNu*9)P6Nvfrp$OhQM+1@zleu9p;5a9jnZf 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(Recall that the asymptotic bias of an estimator is +its probability limit minus the true parameter.) Can you determine +unambiguously whether the 2SLS estimator tends to underestimate or +overestimate the parameter $β_1$ ? If so, give explanations how. diff --git a/site/problemsets/07/ps07_2022.qmd b/site/problemsets/07/ps07_2022.qmd new file mode 100644 index 0000000..e51a834 --- /dev/null +++ b/site/problemsets/07/ps07_2022.qmd @@ -0,0 +1,55 @@ +--- +title: "ECON 626: Problem Set 7" +#author: "Paul Schrimpf" +date: 2022-12-06 +bibliography: ../../626.bib +--- + +$$ +\def\Er{{\mathrm{E}}} +\def\En{{\mathbb{En}}} +\def\cov{{\mathrm{Cov}}} +\def\var{{\mathrm{Var}}} +\def\R{{\mathbb{R}}} +\newcommand\norm[1]{\left\lVert#1\right\rVert} +\def\rank{{\mathrm{rank}}} +\newcommand{\inpr}{ \overset{p^*_{\scriptscriptstyle n}}{\longrightarrow}} +\def\inprob{{\,{\buildrel p \over \rightarrow}\,}} +\def\indist{\,{\buildrel d \over \rightarrow}\,} +\DeclareMathOperator*{\plim}{plim} +$$ + +# Problem 1 + + +Consider the following linear regression model such that +$$ +Y_i = β_0 + X_i β_1 + u_i , +$$ +where $X_i$ and $Y_i$ are observed random variables. Let us assume that +$\Er [u_i ] = 0$ but $\cov(X_i , u_i ) \neq 0$. +Suppose that there exists a variable $Z_i$ such that +$\cov(X_i , Z_i ) > 0$ and +$\cov(Z_i , u_i ) > 0$. + +Find the asymptotic bias of the 2SLS estimator of +$\hat{\beta}_1$. (Recall that the asymptotic bias of an estimator is +its probability limit minus the true parameter.) Can you determine +unambiguously whether the 2SLS estimator tends to underestimate or +overestimate the parameter $β_1$ ? If so, give explanations how. + +# Problem 2 + +In the linear model, +$$ +Y_i = \beta_0 + X_i \beta_1 + u_i +$$ +assume that $\Er[u_i] = 0$ and $X_i \in \R^1$. Suppose that $\Er[X_i u_i] \neq 0$, but, +somewhat strangely, you assume $\Er[u_i^2|X_i] = \sigma^2$. + +1. Show that a set of two elements that contains $\beta_1$ is identified. Denote this set by $B_1$. +**Hint: use the moment condition $\Er[u_i^2 (X_i-\Er[X_i])]$.** + +2. Describe an estimator for $B_1$ and show that it is consistent. State any additional assumptions needed. + +3. Find the asymptotic distribution of your estimator for $B_1$. State any additional assumptions needed.